Global optimization in least squares problems in FTIR spectroscopy and X-ray crystallography

Least squares problems arise in numerous applications in science and engineering that call for the accurate determination of model parameters that best fit experimental data. The main goal of this thesis is the development of novel optimization approaches to solve important crystal structure determination problems from X-ray diffraction data. Least squares approaches have been proposed for these problems in the past. Towards our goal, we study a number of least squares problems.; In the first part of the thesis, we experiment with a recently developed global optimization algorithm in order to assess the potential of this algorithm for solving least squares problems. We use a large collection of problems from the statistics literature for this purpose. We find new solutions to some of these problems despite their small size and long history.; In the second part of this thesis, we address the problem of simultaneous model identification and parameter estimation in FTIR spectroscopy. First, we show that the current decomposition approaches are not sufficient to provide the best possible results to this problem. Then, we develop the first global optimization approach to the problem and demonstrate that it provides better results compared to existing approaches.; Finally, the last part of the thesis considers the “phase problem” in X-ray crystallography. In particular, we address the problem of estimating the three-dimensional atomic positions of crystal structures from diffraction measurements alone. Starting from the “minimal principle” model, we develop novel optimization formulations and algorithms that exploit the special model structure of the problem. For the case of centrosymmetric structures, we first formulate the problem as a 0–1 linear programming problem and suggest a branch-and-bound algorithm for solving it. Based on empirical observations regarding the nature of the solutions of this model, we then propose to solve it through a system of linear equations for which we develop a fast Gaussian elimination algorithm. For noncentrosymmetric structures, we reduce the phase problem to a mixed-integer quadratic problem and utilize a branch-and-bound algorithm for its solution.